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“IF-THEN” AS a VERSION of “IMPLIES” Draft of July 26, 2021 Matheus Silva “IF-THEN” AS A VERSION OF “IMPLIES” Draft of July 26, 2021 Matheus Silva ABSTRACT Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey of basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted “if-then” as a version of “implies” and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of “if-then” are used instead of being mentioned as the premise and the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction “material conditional” instead of “material implication”. Quine’s interpretation on the topic became hugely influential, if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of “if-then” are mentioned and not used; (2) Quine’s belief that the components of “if-then” are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) the Philonian construction remains counter-intuitive even if it is called “material conditional”; (4) the Philonian construction is more plausible when it is interpreted as a material implication. Keywords: material implication; conditionals; if-then; use-mention fallacy; conditional-assertion theories; Principia Mathematica. 1. INTRODUCTION “much confusion has been produced in logic by the attempt to identify conditional statements with expressions of entailment.” – William & Martha Kneale, The Development of Logic “whereas there is much to be said for the material conditional as a version of “if-then”, there is nothing to be said for it as a version of ‘implies’.” – W. V. O. Quine, Word and Object In the Principia Mathematica, Russell employed the notion of material implication to interpret “if-then” constructions of natural language. According to Quine, this choice of terminology was fallacious, since it involves a confusion between the use and mention of words. Quine’s accusation became influential. This paper will argue that this widely accepted accusation is unfounded, for the antecedent and consequent of “if-then” are mentioned, not used. It is also argued that interpreting conditionals as assertions of material implication can provide fruitful solutions to known puzzles in the literature. It is important to notice, however, that while there’s an interesting proposal to be made and textual evidence that may justify Russell’s choice of terminology, a full-blown defense of material implication will require concepts and intuitions that were completely alien to Russell. For instance, a defense of “if-then” as an assertion of material implication will require modal intuitions that he openly refused in his posthumously published paper “Necessity And Possibility” (1905). According to Russell, the modal operators of “necessity” and “possibility” have only an epistemological or psychological significance and should have no place in formal logic. Instead, Russell tried to deflect the criticisms against material implication with a pragmatic defense of his choice of terminology. The position advanced in this paper couldn’t be more different even if it is inspired in Russell’s writings. Oddly enough, the notion of material implication that is currently perceived as an ancient artefact from the old days can only be reinvigorated into its full force with the use of contemporary ideas that weren’t popular in Russell’s time. 2. THE PRINCIPIA CONTROVERSY 1 Russell’s changed his ideas about logic constantly, but some core views remain the same throughout his lifework1. Russell firmly believed that symbolic logic captures the essence of deductive reasoning and that we should develop a symbolic logic capable of showing that mathematics is reducible to logic. More importantly, he endorsed the notion of implication as fundamental to our understanding of deduction and believed that there are two types of implication: material and formal. Material implication is a proposition which displays a relation between two propositions, let’s say, p and q. The statement “p materially implies q” is symbolized as p ⊃ q and is true unless p is true and q is false, i.e., whenever p is not true or q is true2. Russell interprets “if-then” sentences as assertions of material implication, so “p materially implies q” can also be read as “if p, then q”3. Formal implication is the implication we find today in first order predicate calculus in such formulas as (x) (Fx ⊃ Gx)4. From his discussion of material implication, Russell draws three curious inferences which would be known as the paradoxes of material implication: (1) for any two propositions, one of these propositions must imply the other; (2) false propositions imply all propositions; (3) true propositions are implied by all propositions. These counter-intuitive consequences were bombarded with criticisms. C. I. Lewis was its main detractor5. In The Calculus of Strict Implication, Lewis objected that material implication didn’t do justice to our intuitions about implication: If ‘p implies q’ means only ‘it is false that p is true and q false,’ then the implication relation is far too ubiquitous to be of any use6. The idea that material implication “is far too ubiquitous to be of any use” is motivated by Lewis’ view that p can only imply q when q is a logical consequence of p. In other words, the notion of implication is linked with the notion of logical consequence and its related cousins (“logical inference”, “entailment”, “valid deduction”, etc.). In Interesting theorems in symbolic logic, Lewis drew the apocalyptic consequences from treating implication, and, therefore, logical consequence, as material implication. This meant that the Principia theorems were be under suspicion and symbolic logic would collapse: The consequences of this difference between the ‘implies’ of the algebra and the ‘implies’ of valid inference are most serious. Not only does the calculus of implication contain false theorems, but all its theorems are not proved. For the theorems of the system are implied by the postulates in the sense of ‘implies’ which the system uses. The postulates have not been shown to imply any of the theorems except in this arbitrary sense. Hence, it has not been demonstrated that the theorems can be inferred from the postulates, even if all the postulates are granted. The assumptions, e. g., of ‘Principia Mathematica,’ imply the theorems in the same sense that a false proposition implies anything, or the first half of any of the above theorems implies the last half7. Lewis’ point is that Russell identifies the deductibility of q from p with the material implication of q from p. This implies that in order for q being deducible from p it is enough that p is false or q is true. But this is unacceptable. Given that the proposition “Pigs fly” is false, I’m not willing to admit that every proposition is inferable from “Pigs fly”. If any true proposition is implied by any proposition, and necessarily true propositions are implied by any proposition, it follows that every true proposition is necessarily true. If the proofs of Principia were made in this way, they would not be truths, since a proof is based on premises that are assumed as true in order to arrive at the truth of a conclusion whose truth was not admitted. One of Lewis criticisms is that the notion of material implication ignores modal distinctions that are intuitively tied to implication. To use our contemporary idiolect, a relation of material implication would only 1 Russell’s views about logic are presented in works that are too numerous to mention. Some of the main references include “The Principles of Mathematics” (1903), “The Theory of Implication” (1906), ‘‘If’ And ‘Imply’, A Reply To Mr. MacColl” (1908), “Principia Mathematica” (1910), “Some Explanations in Reply to Mr. Bradley” (1910), “The Philosophical Importance of Mathematical Logic” (1913) and “Introduction to Mathematical Philosophy” (1919). Two articles that were published posthumously, “Recent Italian Work on The Foundation of Mathematics” (1901) and “Necessity and Possibility” (1905), repeat some of the main ideas of his other works. 2 Russell & Whitehead (1910: 7). 3 Russell & Whitehead (1910: 208). 4 Although Russell confusedly thought that formal implication belongs in the propositional calculus. 5 For an overview of the clash between Russell and Lewis, see Barker (2006). 6 Lewis (1914: 246). 7 Lewis (1913: 242). 2 require certain combinations of truth values in the actual world, but logical inference requires a stronger connection: Material implication it will appear, applies to any world in which the all-possible is the real, and cannot apply to a world in which there is a difference between real and possible, between false and absurd. Strict implication has a wider range of application. Most importantly it admits of the distinction of true and necessary, of false and meaningless8. In Symbolic Logic, Lewis and Langford proposed an alternative logical system based on a different notion of implication that would better represent Lewis’ intuitions about the subject: It appears that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible. It fails to hold when valid deduction is not possible9. The relation where p strictly implies q would be symbolized as p ⥽ q, and it would only be true when q is logically inferable from p, and is logically equivalent to ¬◊(p&¬q). The possibility must be understood as a logical possibility, since ◊p means “p is self-consistent”10.
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